Let \(A^s_{p,q} (\mathbb R^n)\) with \(A\in \{B,F\}\) be the nowadays well-known scales of function spaces on \(\mathbb R^n\), where \(s\in \mathbb R\) and \(0<p,q \leq \infty\) (\(p<\infty\) for \(F\)-spaces), and let (essentially) \[ \tilde{A}^s_{p,q} (\Omega) = \big\{ f \in A^s_{p,q} (\mathbb R^n): \text{supp } f \subset \bar{\Omega} \big\}, \] where \(\Omega\) is an arbitrary domain in \(\mathbb R^n\). If \(\Omega\) is bounded, then it is well known under which conditions the embedding \[ \tilde{A}^{s_1}_{p_1, q_1} (\Omega) \hookrightarrow \tilde{A}^{s_2}_{p_2, q_2} (\Omega) \] is compact and what can be said about the degree of compactness expressed in terms of the entropy numbers \(e_k\) of this embedding. It is the main aim of this paper to extend this theory to a class of quasi-bounded domains \(\Omega\) subject to mild regularity assumptions (uniform porosity). The quasi-boundedness is expressed by a box-packing number \(b(\Omega) \geq n\). Under suitable restrictions for the parameters, the authors prove in Theorem 4.2 that \[ e_k \big( \tilde{A}^{s_1}_{p_1,q_1} (\Omega) \hookrightarrow \tilde{A}^{s_2}_{p_2, q_2} (\Omega) \big) \sim k^{-\gamma}, \quad k\in \mathbb N, \] with \[ \gamma = \frac{s_1 - s_2}{b(\Omega)} + \frac{b(\Omega)-n}{b(\Omega)} \left( \frac{1}{p_1} - \frac{1}{p_2}\right), \] where (in particular) \[ s_1 - s_2 > n \left( \frac{1}{p_1} - \frac{1}{p_2} \right) + b(\Omega) \left( \frac{1}{p_2} - \frac{1}{p_1} \right)_+. \] The authors rely on wavelet characterizations of the corresponding spaces. Then the embedding between the above function spaces can be reduced to embeddings between related sequence spaces. The assertions are applied to discuss the distribution of eigenvalues of related elliptic differential operators.